Papers

Let $(M^n,g_0)$ be a closed hyperbolic manifold, $n\geq3$. We compute the Hessian of volume entropy at $g_0$ under the constraint $Sc_g=-n(n-1)$ and express it as a quadratic form on transverse-traceless symmetric two-tensors. In dimension three, the Hessian is negative definite, and every sufficiently $C^\infty$-close metric with $Sc_g\geq-6$ has entropy at most $2$, with equality precisely for pullbacks of $g_0$. In dimensions $n\geq4$, a universal spectral threshold determines the Hessian sign. Nonzero trace-free Codazzi tensors give positive directions, and hyperbolic bending provides smooth constrained entropy saddles in every such dimension.

Sasakian manifolds, a subclass of contact metric manifolds, play a central role in modern differential geometry and mathematical physics, where they arise naturally in geometric mechanics, CR geometry, and supersymmetric field theories. The weak contact metric (w.c.m.) structure generalizes the contact metric structure and provides a broader framework for studying contact geometry and its applications. A fundamental problem is to understand how curvature conditions constrain the underlying contact metric structure and, in particular, how they distinguish Sasakian geometry from its weaker variants. Using the partial Ricci flow, we characterize Sasakian structure (among w.c.m. manifolds) under the $(\kappa,\mu)$-nullity condition related to curvature. For $\kappa<1$ we find conditions under which a w.c.m. manifold admits a bi-Legendrian structure, and for $\kappa=\mu=0$ establish splitting and classification results.

We construct Golod–Shafarevich towers of arithmetic surfaces over the integers. In particular, we obtain an arithmetic surface with a section and infinite geometric etale fundamental group, answering a question raised by Bost and Charles about the existence of such surfaces. Taking generic fibres gives curves over the rationals with infinite geometric etale towers in which a rational point splits completely. This answers questions posed by Ihara and by Frey, Kani and Völklein. Taking the generic fibres of the tower, we also obtain sequence of curves over the rationals whose genera goes to infinity and whose logarithmic conductors grows no more than linearly in the genera, answering a question asked by Venkatesh at PCMI in 2022. The authors worked together on this problem since 2022 and developed a general strategy. They were able to bring it to fruit only after a collaboration with GPT5.6 Sol, Fable and Aletheia a Gemini-powered internal agent developed at Google DeepMind.

We provide a complete study of the optimal control problem associated to the so-called Dyson equation, which describes the mean field evolution of the spectrum of large random matrices. We treat both the case of regular and singular terminal data. We construct admissible controls and prove the existence of optimal ones. We then proceed to give a thorough mathematical analysis of the associated Hamilton-Jacobi-Bellman equation by means of the theory of viscosity solutions in the space of measures. In particular, we characterize the value function of the optimal control problem as the unique viscosity solution to this infinite dimensional HJB equation.

We present an algorithm for handling the small cell problem in Cartesian cut cell grids. The advantage of this variation is that it preserves the base scheme's steady state exactly. We show several computational examples using finite volume schemes in two space dimensions.

In this note, we show that over a geometrically irreducible smooth projective curve over an infinite field $k$, the moduli stack of semistable vector bundles of fixed determinant is $\mathbb{A}^1$-connected if and only if the moduli stack admits a $k$-rational point. For this we use Langton's elementary modifications for vector bundles at the level of families. In addition, we prove the $\mathbb{A}^1$-connectedness of the moduli stack of symplectic bundles with forms valued in a fixed line bundle $L$ on a smooth projective curve of genus $g \ge 2$ over an infinite field $k$ with $C(k)\neq \emptyset$. As an application, we deduce the $\mathbb{A}^1$-connectedness of moduli stack of quasi-parabolic symplectic vector bundles with forms valued in a fixed line bundle.

We study a one dimensional nonlinear Klein-Gordon equation with a potential, a localized quadratic nonlinearity, and a non-localized quartic nonlinearity. We assume that the linearized operator has a single discrete eigenvalue below the continuous spectrum, corresponding to an ”internal mode”. This eigenvalue generates localized, time-periodic solutions for the linear equation. Assuming the natural Fermi Golden Rule, we give a global description of the dynamics of the amplitude of the internal mode through the radiation damping mechanism, and prove scattering for the radiation. Our approach is based on the distorted Fourier transform and a collection of refined dispersive decay and smoothing estimates. The corresponding cubic problem is closely related to the question of global-in-space dispersive asymptotics for perturbations of kink solutions in the classical $\phi^4$ model, which remains a challenging open question. From the perspective of nonlinear estimates, a quartic (non-localized) interaction is essentially sharp for our analysis, and closing the estimates with such a low degree nonlinear term requires a careful and delicate treatment. As a direct application of our analysis, we prove a result on the asymptotic stability of kinks for classes of scalar field models that can be seen as perturbations of the $\phi^4$ model.

We prove a noncommutative maximal ergodic inequality for averages along polynomial sequences. Let $\gamma$ be a trace-preserving automorphism of a semifinite von Neumann algebra $(\mathcal N,\tau)$. We show that the associated polynomial averages \begin{equation*} A_Nf:=\frac1N\sum_{n=1}^N\gamma^{P(n)}(f), \qquad N\in\mathbb N, \end{equation*} satisfy a strong maximal inequality on $L_p(\mathcal N)$ for every $1<p<\infty$, extending the previously known restricted range of $p$. The proof follows Bourgain's major-arc strategy but requires substantially new ideas and tools for operators that may have further applications in noncommutative analysis. More precisely, we obtain a localized maximal inequality by developing a noncommutative version of Stein's extrapolation, novel combinatorial methods, and a surprising multilinear version of Doob's maximal inequality. For the required decaying $L_2$-approximation, we combine two of Bourgain's constructions in a way that avoids the multi-frequency maximal inequality used in the scalar proof.

Let $q=2^n$ with $n\ge6$, and let $\mathbb{F}_q$ be the finite field with $q$ elements. For a subspace $E$ of $\mathbb{F}_q$, define \[ S(E)=\sum_{x\in E\setminus\{0\}}x^{-1}.\] Let $N_{n,k}$ denote the number of $k$-dimensional $\mathbb{F}_2$-subspaces $E$ for which $S(E)=0$. We prove that, if $k\ge3$ and $n\ge2k+1$, then \[ \left|\frac{2^nN_{n,k}}{\genfrac{[}{]}{0pt}{}{n}{k}_{2}}-1\right|<2^{r^2+r}2^{n(1-r/2)},\] where $r=\lfloor\frac{k-1}{2}\rfloor$, and $\genfrac{[}{]}{0pt}{}{n}{k}_{2}$ is the Gaussian binomial coefficient. In particular, \[ N_{n,k}\sim2^{-n}\genfrac{[}{]}{0pt}{}{n}{k}_{2}\quad (n\to\infty)\] uniformly for $n/3<k\le(n-1)/2$. Combined with the known low-dimensional cases and the symmetry between dimensions $k$ and $n-k$, and the elementary middle-dimensional subfield construction, this proves a conjecture of Carlet: for every $3\le k\le n-3$, there exists a $k$-dimensional $\mathbb{F}_2$-subspace $E$ such that $S(E)=0$.

We study torus surgery on the Cartwright–Steger surface and the degree-two cover construction proposed in \cite[Remark~1]{ASY}. Every torus in this surface is integrally nullhomologous, and surgery on a fixed disjoint collection of such tori cannot decrease the first Betti number. We then determine the nodal-curve subgroup and complete the fundamental-group calculation in the degree-two construction. The resulting minimal symplectic manifold has signature zero and fundamental group $\Z/2$; for compatible symplectic gluings, its universal cover is an exotic symplectic $31\CP^2\#31\overline{\CP}^{\,2}$. The construction on the original Cartwright–Steger surface gives a minimal symplectic manifold with signature $-1$ and fundamental group $(\Z/2)^2$.

We consider weak solutions $u \in W^{1,n}(U,\mathbf{R}^{n+1})$ of the $H$-system $- \operatorname{div} ( |\mathrm{D} u|^{n-2} \mathrm{D} u ) = (H \circ u) \, \mathbf{J} u$, where $n \ge 2$, $U$ is the open unit ball in $\mathbf{R}^n$, and $H$ is bounded. We show that $u$ is locally bounded if its distortion $|\mathrm{D} u|^n / |\mathbf{J} u|$ is bounded on the set where $|u|$ is large. By a result of Fusco, Kristensen, Leone, and Verde, it follows that $u$ is locally Hölder continuous if $H$ is continuous, and continuously differentiable with locally Hölder continuous derivative if $H$ is locally Lipschitz. The proof pushes forward $|\mathrm{D} u|^{n-2} \mathrm{D} u \circ \mathrm{D} u^*$ by $u$ to a linear functional on continuous compactly supported maps of $\mathbf{R}^{n+1}$ into $\operatorname{Hom}(\mathbf{R}^{n+1},\mathbf{R}^{n+1})$, whose values on derivatives are controlled by the equation. When the distortion is bounded, this functional satisfies the hypotheses of an inequality of Michael-Simon type for measures satisfying a first-order partial differential equation, due to De Philippis, Gennaioli, Pigati,and Rindler. The inequality yields lower bounds for the mass ratios of the push-forward of $|\mathrm{D} u|^n \mathscr{L}^n$ by $u$, and these imply boundedness.

We use the breeding technique to construct genus-3 Lefschetz pencils for which the total spaces have $c_1=0$. We compute the fundamental groups of our Lefschetz pencils and identify them with the fundamental groups of certain torus bundles over tori. Using work of Friedl and Vidussi, the total spaces of our pencils are homeomorphic to torus bundles over tori. These bundles do not admit sections.

In this paper, we study closure relations of Simpson strata in Hodge moduli spaces over a smooth complex projective curve of genus $g\geq2$ and prove in every rank that the Heinloth weight strictly increases whenever a specialization changes the fixed component. For stable full chains, we refine this numerical condition by comparing the degrees of the corresponding filtration steps and construct counterexamples to Simpson's nestedness conjecture for the Dolbeault and de Rham moduli spaces in every rank $n\geq4$. Using the maximum property of this weight, we also discuss Simpson filtrations from the viewpoint of $\Theta$ stratifications.

The classical Descartes' rule of signs bounds the number of positive roots of a real polynomial in one variable by the number of sign changes of its coefficients. This bound only depends on the order of the exponents and on the signs of the coefficients, that is, on the oriented matroids of the exponent and coefficient matrices. We give a multivariate generalization of this rule to arbitrary fewnomial systems: for systems of $n$ real polynomial equations in $n$ variables with $n+k+1$ monomials, we obtain upper bounds for the number of nondegenerate positive solutions which only depend on sign conditions on the maximal minors of the coefficient matrix and of the exponent matrix and its liftings. For $k=1$ they are the Descartes rules of signs for circuits of Bihan-Dickenstein and Bihan-Dickenstein-Forsgård. Our approach starts from the Gale duality and the Khovanskii-Rolle argument of Bihan and Sottile, which reduce the problem to counting common zeros of iterated Jacobians in a polyhedron $\Delta$. This polyhedron is a chamber of a hyperplane arrangement whose combinatorics is determined by the oriented matroid of the coefficient matrix. We bound the number of these zeros by the Bézout number minus the zeros forced in the other chambers by a topological degree, which we compute by a facet-count formula involving the signs of minors of a Gale dual matrix of the exponent matrix. This leads to a recursion over the flats of the arrangement, which ends on lines with the Descartes rules for circuits. In this way the coefficients enter the bounds through the combinatorics of the arrangement, and the exponents through sign conditions on minors.

Homogeneous numerical integration (HNI) combines Euler's identity and Stokes' theorem to integrate homogeneous polynomials over a polytope $P$. Existing algorithms return scalar integrals or reusable moment families. Any resulting moment table can be recoded afterward: once all degree-$q$ moments are known, integration on $\mathcal H\_q$ (degree-$q$ forms) has a trivial representation by a pure order-$q$ distribution supported at any single arbitrary point. Our contribution is instead to propagate functionals through the face complex, once for $(P,q)$ and without first forming the moment vector, to obtain a signed vertex-derivative coefficient functional on $\mathcal H\_q$, specific to $P$ and determined directly by its supplied face geometry and chosen anchors. A finite-dimensional Riesz framework describes its nonuniqueness, gives an intrinsic $L^2(P)$ error for restricted formulas, and supports weighted post-processing. The independently generated degree-$q$ and degree-$2q$ coefficient vectors satisfy an explicit compatibility relation. Vertex anchors can prune the propagated representation as shown in some polygon examples and an offline benchmark quantify this effect without claiming global cardinality optimality or improved conditioning. The construction covers convex polytopes and oriented non-convex polyhedral domains in arbitrary dimension.

We give a Seifert-form criterion for upper bounds on topological shaking numbers and prove that isotropic certificates lift under $(m,1)$-cabling.

We study the reflexivity property of ideals in numerical semigroups. In particular, we characterize irreducible reflexive ideals and determine the Apéry set of a reflexive ideal. Moreover, we study how reflexivity allows us to characterize some classes of numerical semigroups and the interaction of reflexivity with the property of being a trace ideal.

We prove the heuristically predicted lower bound for the number of rational points of height at most $Q$ lying within $\epsilon/Q$ of a fixed analytic nondegenerate manifold in $\mathbb{R}^n$, provided that $\epsilon \asymp Q^{-\tau}$ for some $\tau \leq 3/(2m+1)$, where $m$ is the codimension of the manifold. Our result establishes the lower bound well beyond the previously conjectured range $\tau \leq 1/m$, and improves upon a recent result of Schindler, Srivastava, and Technau, who established the same lower bound for $\tau \leq 3/(2n-1)$.

Motivated by Gal's conjecture, Ferroni and Higashitani conjectured that the $h^*$-polynomial of any Gorenstein lattice polytope admitting a quadratic triangulation is $\gamma$-positive. Together with conjectures predicting quadratic properties of toric ideals of smooth lattice polytopes, this suggests that smooth Gorenstein lattice polytopes should have $\gamma$-positive $h^*$-polynomials. In this paper, we prove that the Ehrhart polynomial of the polar dual of every pseudo-symmetric simplicial reflexive polytope is magic positive. For pseudo-symmetric smooth Fano polytopes, we prove the stronger statement that all magic coefficients are strictly positive. Consequently, for every pseudo-symmetric simplicial reflexive polytope, its polar dual is Ehrhart positive and has a real-rooted and $\gamma$-positive $h^*$-polynomial. We also prove that the Ehrhart polynomial of the polar dual of the symmetric edge polytope of every cycle is magic positive. This gives an affirmative answer to a question of Konoike, who had previously proved partial positivity results for this family.

Let $\mathcal H$ be the space of support functions of centrally symmetric, strictly convex, compact planar bodies whose boundary is a $C^1$ curve, endowed with the $C^1$ topology. We prove that for a residual subset of $\mathcal H$ the outer billiard maps about the associated bodies admit unbounded orbits escaping to infinity. In particular, this residual set includes support functions arbitrarily $C^1$-close to those of disks or ellipses. This provides an affirmative answer, in the strictly convex $C^1$ setting, to a famous question of Moser and Neumann concerning the existence of unbounded outer-billiard orbits. The key ingredient is a rigidity theorem: if $h\in\mathcal H$ has a purely singular radius-of-curvature measure, then every continuous invariant tangent graph consists entirely of periodic points. We then show that for a generic table in $\mathcal H$ these periodic invariant graphs are absent, allowing us to construct escaping orbits via Mather's diffusing mechanism in a Birkhoff region of instability.

The performance of stationary iterative methods is memory-bound, due to the low arithmetic intensity. So, we propose a mixed-precision technique to reduce the amount of data movement in matrix-splitting iterative methods. We demonstrate both theoretically and experimentally that this technique will generally converge at a similar rate as an exact-precision iteration, depending on the precision used and the conditioning of the $M$ matrix. Furthermore, we demonstrate that iterative refinement can be used to refine the solution to full double-precision accuracy. Using this approach, we mixed half and double precision to achieve an average speedup over uniform double precision of $1.60\times$ for scalar and block Jacobi and $1.25\times$ for scalar and block Gauss-Seidel. Furthermore, we reduced the precision in the SSOR iteration used by OVERFLOW, a NASA code for compressible fluid dynamics simulations, and achieved a $1.14\times$ speedup to the overall application.

We consider nonlocal divergence-form operators of the type $$PV\int_{\mathbb{R}^n}(u(x)-u(y))K(x, y) dy=0\quad \text{in} \; B_2,$$ where the kernel $K$ is symmetric ($K(x, y) = K(y, x)$). We first show that there exists a kernel satisfying the conditions of \cite[Open Question 2.1]{FRRO24} for which this problem admits a bounded discontinuous weak solution, answering that question negatively. We then impose the stronger one-sided lower bound suggested in \cite{CS20,IS20}, together with an $L^p$ bound of $K$ over annuli, and prove a coercivity estimate and interior Hölder regularity of weak solutions. In particular, this establishes the coercivity conjecture of \cite{CS20,IS20} under the additional $L^p$ hypothesis for $p>1$.

It was recently proved that an embedded minimal annulus with free boundary in the unit ball $\Bbb B^3$ of the Euclidean space is, up to a rotation, the critical catenoid. We prove that an immersed, possibly branched, free boundary minimal annulus in $\Bbb B^3$ with embedded boundary components is necessarily free of branch points and is embedded. This shows the uniqueness of the critical catenoid holds under these weaker hypotheses. We then apply these results to prove that if $\Omega$ is an annulus in $\Bbb S^2$ for which there exists a smooth function satisfying the overdetermined eigenvalue problem \begin{equation*} \begin{cases} \Delta u+ 2u = 0 \quad \text{on} \quad \Omega\\ \qquad \quad u=0 \quad \text{in}\quad\partial\Omega\\ \quad\,\,\,\, |\nabla u|=1 \quad \text{in}\quad \partial\Omega. \end{cases} \end{equation*} then $\Omega$ is, up to rotation, a rotational annulus with equatorial symmetry.

We establish global energy compactness for rough three-dimensional axisymmetric Euler flow without swirl and use it to construct hydrodynamic limits of collisional kinetic equations beyond finite-time Euler regularity breakdown. For relative vorticity in $L^1\cap L^p$, regular approximate Euler flows are globally compact in $C_tL_x^2$ for $p\ge4/3$, without a sign condition or any spatial-moment assumption. Below $4/3$, the same conclusion holds under uniform control of the absolute impulse. We identify the sharp radial dynamics governing this confinement: the absolute impulse is propagated without a sign condition for $p\ge5/3$, while for $1<p<5/3$ it is controlled under one-sided $L^{p^\sharp}$ integrability, where $p^\sharp= {4p} / (3p-1)$, and this exponent is sharp. In particular, for one-sign finite-impulse vorticity the resulting global energy compactness holds throughout the full range $p>1$. For the Coulomb Landau equation and a class of non-cutoff soft-potential Boltzmann equations, we construct strong solutions on lifespans $T^\varepsilon\to\infty$ realizing prescribed rough finite-energy Euler initial data and obtain energy-conserving axisymmetric Euler limits on every fixed time interval. If the corresponding regular Euler evolution loses regularity in finite time, the hydrodynamic fields converge to the regular Euler solution up to the singular time and have a common strong trace there, while subsequences yield global energy-conserving Euler continuations past that time.

We establish that the skew-symmetric bicharacter on the group of $\ell$-weights of a quantum affine algebra is invariant under Chari's braid group action. Furthermore, we define an extension of this bicharacter to the group generated by prefundamental $\ell$-weights and show that this invariance continues to hold under the extended braid group action of Frenkel-Hernandez. As immediate applications, we obtain braid group actions by automorphisms on the quantum tori associated respectively with the category of finite-dimensional representations of the quantum affine algebra and with the category $\mathcal{O}^{\mathfrak{b}}$ of its Borel subalgebra. In addition, we derive explicit formulas for quantum $QQ$-systems in all simply-laced types. Finally, we show that our extended bicharacter yields a quantization matrix compatible with the Hernandez-Leclerc cluster algebra categorified by $\mathcal{O}^{\mathfrak{b}}$ for all finite Dynkin types, extending Bittmann's construction in the simply-laced setting.

Let $d\geq2$ be an integer, and let $\operatorname{MPoly}^d$ be the moduli space of degree-$d$ polynomials. For every number field $K$, we prove that there exists $\epsilon_{K,d}>0$ such that \[ \left\lbrace\alpha\in\operatorname{MPoly}^d(K^{\mathrm{ab}})\colon h_{\mathrm{crit}}(\alpha)<\epsilon_{K,d}\right\rbrace=\left\lbrace\alpha\in\operatorname{MPoly}^d(K^{\mathrm{ab}})\colon h_{\mathrm{crit}}(\alpha)=0\right\rbrace \] is finite, where $h_{\mathrm{crit}}$ is the critical height. In particular, only finitely many $K^{\mathrm{ab}}$-rational points of $\operatorname{MPoly}^d$ are postcritically finite. The proof combines the correspondence method of Ji–Song–Xie with local ramification estimates and the uniform finiteness of intertwined classes in $\operatorname{MPoly}^d$. We also prove an analogous finiteness theorem for non-preperiodic marked points in non-isotrivial one-parameter polynomial families.

We study the position and shape of invasion fronts for spatially periodic Fisher–KPP equations on higher-dimensional lattices, starting from localized initial data. Periodic spectral geometry determines a directional logarithmic upper bound. When the linear growth rate is independent of the propagation coordinate, we prove the matching axial Bramson correction and identify the critical pulsating front approached by the solution. The selected profile includes the local periodic cell and a bounded phase whose derivative tends to zero. This description also determines the times at which prescribed levels reach distant lattice sites and the asymptotic time gaps between levels or neighboring cells. The proof combines a periodic transverse heat kernel, a scalar lower comparison, and critical-tail analysis. For growth rates varying in every coordinate, a positive Dirichlet quotient comparison reduces the matching lower bound to two explicit estimates on the linear seed and its accumulated nonlinear loss.

A pseudo-Riemannian manifold is called geodesic orbit (GO) if every geodesic is an orbit of a one-parameter group of isometries. For Riemannian $H$-type groups, the GO property was completely characterized by A. Kaplan and C. Riehm in terms of the dimension of the centre and the structure of the underlying Clifford module. We solve the corresponding problem for pseudo $H$-type groups $N_{r,s}$, that is, 2-step nilpotent Lie groups whose Lie algebra $\mathfrak n_{r,s}=\mathfrak z\oplus\mathfrak v$ is built from an admissible module $\mathfrak v$ of the Clifford algebra $\mathrm{Cl}(\mathbb R^{r,s})$, endowed with the left-invariant metric whose restriction to the centre is indefinite ($s\ge 1$). We prove that $N_{r,s}$ is naturally reductive if and only if $(r,s)\in\{(0,1),(1,2)\}$, and then it is GO for every admissible module. We also prove that $N_{3,4}$ is GO, but not naturally reductive, exactly when $\mathfrak v$ is a minimal admissible module, and that in all remaining cases $N_{r,s}$ is not GO. Thus, in contrast with the Riemannian situation, where a seven-dimensional centre admits GO metrics on isotypic modules of dimensions 8, 16 and 24, the exceptional pseudo-Riemannian case is GO only for the smallest possible module. For $N_{3,4}(\mathfrak v_{\min})$ we explicitly construct the one-parameter isometry groups generating all geodesics, including the null ones, and describe the affine family of their generators.

We study empirical processes of observable extreme generators in stationary Poisson-Laguerre tessellations, with each generator weighted either by one or by the volume of its cell. Under moment assumptions that allow heavy-tailed marks, we prove functional central limit theorems as the observation window grows, on compact intervals where the mark law has a bounded density. We also obtain covariance asymptotics with a surface-order error and quantitative univariate and multivariate Gaussian approximation.

We study higher moments of quantum fluctuations and microscopic correlation energy in the three-dimensional Gross–Pitaevskii regime. For a smooth, compactly supported, nonnegative radial interaction and an $N$-independent normalized initial condensate wave function in $H^6$, we propagate the third spectral moment of a positive correlated energy along the exact many-body evolution. A uniform initial bound controls the third moment of the depletion number and sixth moments of collective one-particle fluctuations on every compact time interval, independently of $N$. For a nonzero interaction and a smooth, compactly supported initial condensate, we also identify the excitation-energy measure at scale $N^2$, weighted by the first energy. This limit holds for norm-convergent pure-state families with a uniform bound on the initial third correlated-energy moment. Its positive-energy part is a spectral measure of the relative scattering operator $-2\Delta+v$, multiplied by a time-dependent dressed contact. We compute the radial density and prove strict positivity of the contact for nonzero references with finite kinetic energy and finite mean excitation number. The same evolving states have bounded physical fluctuation moments and excitation-energy moments of order $p>1$ bounded below by a positive multiple of $N^{2p-2}$, in the extended spectral sense. Inverse powers of the positive initial energy construct admissible pure and mixed families. For the norm-convergent pure-state class, we determine the scaled interaction-energy moments for $1\le p<2$ and bound the second moment by $O(N^2)$. A finite cubic approximation has norm error $O(N^{-3/16})$ on the energy form domain and $O(N^{-5/4})$ on a stronger initial domain of its generator.